Talk:Proper convex function

Based on the definition of "proper", it ensures that for inf(sup) problem, the lower/upper bound always exists.

Original Research
Rockafellar is given as the source for this article. There is undue emphasis on field-properties and a reversion of the point of convex analyis in the following OR: This definition takes account of the fact that the extended real number line does not constitute a field because, for example, the value of the expression &infin; &minus; &infin; is left undefined.

It is always possible to consider the restriction of a proper convex function f to its effective domain



\mbox{dom} f = \left\{x : f(x) < \infty \right\} $$

instead of f itself, thereby avoiding some minor technicalities that may otherwise arise. The effective domain of a convex function is always a convex set. On the contrary, ERV functions are used for representing constraints, and provide modelling advantages, particularly for stochastic programming (see Rockafellar's 1993 Von Neumann lecture in SIAM Review).

The OR statements preferring restrictions to the effective domain are unsourced and erroneous. Using the convexity of the epigraph as the definition avoids the technicalities of bad definitions. Kiefer .Wolfowitz 19:05, 25 April 2011 (UTC)


 * I believe I have dealt with these issues by both removing the questionable statements or giving additional sources. Zfeinst (talk) 22:15, 19 October 2011 (UTC)

minima of proper convex functions
Shouldn't the "properties" section discuss minimization over a convex set? Isn't that often why you care that a function is proper convex (e.g., Fenchel duality)?

I am thinking of statements like those on page 4 of. There are other useful facts later in that handout, e.g., about recession cones. Eclecticos (talk) 18:54, 2 November 2015 (UTC)

Domain and effective domain
The notation for domain ($$\mbox{domain}(f)$$) and effective domain ($$\mbox{dom}(f)$$) can be easily confused by a novice. Whenever possible I explicitly use $$X$$ instead of domain. — Preceding unsigned comment added by David Pal (talk • contribs) 22:59, 8 December 2023 (UTC)